A concept of discrete space-time was first suggested by V. Ambartsumian and D. Ivanenko in 1930. This concept has not been formulated in a strict mathematical form, but often is related to a hypothesis of a minimal fundamental length, e. g., in non-linear field theories with self-interaction constants of dimension of length. One usually considers a lattice in coordinate space and introduces a cut-off factor in a momentum space.
An acceptable mathematical formalization of a discrete space-time can be given in terms of topological spaces Y where a connected component of any point y of Y is the closure of y. For instance, if Y is a Hausdorff space, then a connected component of its point y is this point itself, i. e., Y is a totally disconnected space.
This is the case of a discrete topological space, the space of rational numbers, vector spaces and analytic manifolds over fields with ultra-metric absolute values. In algebraic quantum field theory, the spectrum of some C*-algebras, e. g., the C*-algebras of probability measures, is totally disconnected.
G. Sardanashvily, Discrete space-time, Enciclopaedia of Mathematics (Springer)
D.Ivanenko, G.Sardanashvily, Towards a model of discrete space-time, Russ. Phys. J. 21 (1978) 1508.